Convexity of reflective submanifolds in symmetric \(R\)-spaces
A reflective submanifold of a Riemannian manifold \(P\) is a connected component of the fixed point set of an involutive isometry of \(P\). In this paper, the authors prove a theorem stating that a reflective submanifold \(M\) of a symmetric \(R\)-space \(P\) is convex: any shortest geodesic arc in \(M\) is still shortest in the ambient space \(P\). Moreover, the authors provide explicit counterexamples to this statement for symmetric ambient spaces which are not symmetric \(R\)-spaces, namely for a flat \(2\)-torus with non-rectangular rhombic lattice (the long diagonal is reflective, but not convex), and the space \({\mathrm{Ad}}({\mathrm{SU}}(3))\), i.e., the quotient of \({\mathrm{SU}}(3)\) by its center. Symmetric \(R\)-spaces are defined as particular orbits of \(s\)-representations, but the proof of the theorem uses the following intrinsic characterization, proven by \textit{O. Loos} [Math. Z. 189, 211--226 (1985; Zbl 0583.53044)]: Given a maximal Abelian subspace \({\mathfrak t}\) of the Lie triple system associated to \(P\), the unit lattice of \(P\) is defined as \(\{X\in {\mathfrak t}\mid {\mathrm{Exp}}(X)=o\}\), where \(o\) is the chosen basepoint of \(P\); for a symmetric \(R\)-space, this unit lattice permits a basis over \({\mathbb Z}\) which is orthogonal. This fact is used to show that the tangent cut locus of the reflective submanifold \(M\) at \(o\in M\) is given by the intersection of \(T_oM\) with the tangent cut locus of \(P\) at \(o\), which is equivalent to the assertion of the theorem.
scientific article; zbMATH DE number 6141568
scientific article